The PM’s deal – which covers the terms of the UK’s divorce and the framework of future relations with the EU – has already been agreed with EU leaders. But it needs to pass a vote by MPs before it is accepted.

The UK is due to leave the EU on 29 March 2019 whether the deal is passed by MPs or not.

Mrs May’s deal is facing opposition from many of her own MPs, as well as Labour and other opposition parties including the Remain-supporting Liberal Democrats.

The DUP – which Mrs May’s Conservative Party relies on for a majority in Parliament – has said it will not back the deal.

But Brexit minister Kwasi Kwarteng dismissed suggestions that the government had accepted it would lose next week’s vote and was planning on returning to Brussels.

“The plan is to win the vote on Tuesday, or whenever it comes,” Mr Kwarteng told BBC Radio 4’s Today programme.

He said a week was “a very long time in politics” and he was “very hopeful” the deal would be voted through.

BBC assistant political editor Norman Smith said reassurances on the Irish backstop were likely to include proposals to minimise any regulatory differences between Northern Ireland and the rest of the UK.

Plans to give Stormont a role in deciding whether the backstop should come into force were also likely.

There could be further possible safeguards for Parliament, with MPs perhaps being given a vote before the UK enters the backstop and the right to notify Brussels of the UK’s intention to quit the backstop within a specified time period, our correspondent added.

Government sources also said they hope to set out further reassurances from the EU that the backstop is only temporary.

Meanwhile, writing in Daily Telegraph, ex-foreign secretary Boris Johnson said the option of leaving the EU with no deal is “closest to what people actually voted for” in the 2016 EU referendum.

And Tory MP Damian Green – also an ex-cabinet minister – said the onus was on the MPs to say what deal they would support.

‘United on one thing’

Tory Dame Caroline Spelman, who organised the MPs’ letter with Labour MP Jack Dromey, said “crashing out” of the EU without a deal would cause job losses.

Dame Caroline – a Remain supporter who was environment secretary for two years when David Cameron was prime minister – told BBC Radio 4’s Westminster Hour programme that 209 MPs had signed the letter.

Asked if the prime minister “gets it”, Dame Caroline said: “Yes, I definitely think she gets it. She wouldn’t have invited us to come in and see her if she didn’t.”

Dame Caroline said the signatories to her letter included Brexit and Remain supporters – but the letter did not bind them to supporting the PM’s withdrawal deal.

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Instead, Dame Caroline said, it creates a “platform” which would “stabilise the economy and give reassurance to manufacturing”.

“We are united on one thing – we want to protect jobs and livelihoods by making sure we don’t crash out without a deal,” she said.

The MPs have been invited to meet the prime minister on Tuesday.

Boris: ‘No deal is popular’

Many Conservative MPs continue to believe the deal does not represent the Brexit the country voted for, and some are actively calling for Britain to leave with no deal.

If the UK leaves the EU without a deal, it would automatically fall back on World Trade Organization rules – which would apply automatically to trade between the UK and EU.

Writing in the Telegraph on Monday, Brexiteer Mr Johnson said of all the options suggested, the no-deal option is “gaining in popularity” and dismissed the warnings against it which he said were “downright apocalyptic”.

Mr Johnson said he wants Mrs May to remove the backstop from the withdrawal agreement, “to give real legal protection to the UK”.

“Failing that, we should approach the challenge of leaving on WTO terms in a way that is realistic and sensible,” he said.

On Sunday, Tory MP Peter Bone told Sky News the best way to “get on” with Brexit was to leave without a deal – which would be “absolutely OK”.

He said support for leaving without a deal was “hardening”.

But speaking on Sunday, Mrs May warned that if Parliament rejects her Brexit deal, the country faces “uncharted territory”.

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The UK’s exit in March was “in danger” if MPs did not vote for it, she added.

As well as the invite to all signatories of the letter to Downing Street, Mrs May has also invited all Tory MPs to drinks receptions on Monday and Wednesday.